3.552 \(\int \frac{x^2}{(c+a^2 c x^2)^2 \tan ^{-1}(a x)^2} \, dx\)

Optimal. Leaf size=43 \[ \frac{\text{Si}\left (2 \tan ^{-1}(a x)\right )}{a^3 c^2}-\frac{x^2}{a c^2 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)} \]

[Out]

-(x^2/(a*c^2*(1 + a^2*x^2)*ArcTan[a*x])) + SinIntegral[2*ArcTan[a*x]]/(a^3*c^2)

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Rubi [A]  time = 0.138606, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {4942, 4970, 4406, 12, 3299} \[ \frac{\text{Si}\left (2 \tan ^{-1}(a x)\right )}{a^3 c^2}-\frac{x^2}{a c^2 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)} \]

Antiderivative was successfully verified.

[In]

Int[x^2/((c + a^2*c*x^2)^2*ArcTan[a*x]^2),x]

[Out]

-(x^2/(a*c^2*(1 + a^2*x^2)*ArcTan[a*x])) + SinIntegral[2*ArcTan[a*x]]/(a^3*c^2)

Rule 4942

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> Simp[
((f*x)^m*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^(p + 1))/(b*c*d*(p + 1)), x] - Dist[(f*m)/(b*c*(p + 1)), Int[
(f*x)^(m - 1)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[e
, c^2*d] && EqQ[m + 2*q + 2, 0] && LtQ[p, -1]

Rule 4970

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Dist[d^q/c^(m
+ 1), Subst[Int[((a + b*x)^p*Sin[x]^m)/Cos[x]^(m + 2*(q + 1)), x], x, ArcTan[c*x]], x] /; FreeQ[{a, b, c, d,
e, p}, x] && EqQ[e, c^2*d] && IGtQ[m, 0] && ILtQ[m + 2*q + 1, 0] && (IntegerQ[q] || GtQ[d, 0])

Rule 4406

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(c + d*x)^m, Sin[a + b*x]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0]
&& IGtQ[p, 0]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 3299

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

Rubi steps

\begin{align*} \int \frac{x^2}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2} \, dx &=-\frac{x^2}{a c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\frac{2 \int \frac{x}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)} \, dx}{a}\\ &=-\frac{x^2}{a c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\frac{2 \operatorname{Subst}\left (\int \frac{\cos (x) \sin (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{a^3 c^2}\\ &=-\frac{x^2}{a c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\frac{2 \operatorname{Subst}\left (\int \frac{\sin (2 x)}{2 x} \, dx,x,\tan ^{-1}(a x)\right )}{a^3 c^2}\\ &=-\frac{x^2}{a c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\frac{\operatorname{Subst}\left (\int \frac{\sin (2 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{a^3 c^2}\\ &=-\frac{x^2}{a c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\frac{\text{Si}\left (2 \tan ^{-1}(a x)\right )}{a^3 c^2}\\ \end{align*}

Mathematica [A]  time = 0.109486, size = 40, normalized size = 0.93 \[ \frac{\text{Si}\left (2 \tan ^{-1}(a x)\right )-\frac{a^2 x^2}{\left (a^2 x^2+1\right ) \tan ^{-1}(a x)}}{a^3 c^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/((c + a^2*c*x^2)^2*ArcTan[a*x]^2),x]

[Out]

(-((a^2*x^2)/((1 + a^2*x^2)*ArcTan[a*x])) + SinIntegral[2*ArcTan[a*x]])/(a^3*c^2)

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Maple [A]  time = 0.065, size = 37, normalized size = 0.9 \begin{align*}{\frac{2\,{\it Si} \left ( 2\,\arctan \left ( ax \right ) \right ) \arctan \left ( ax \right ) +\cos \left ( 2\,\arctan \left ( ax \right ) \right ) -1}{2\,{a}^{3}{c}^{2}\arctan \left ( ax \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a^2*c*x^2+c)^2/arctan(a*x)^2,x)

[Out]

1/2/a^3/c^2*(2*Si(2*arctan(a*x))*arctan(a*x)+cos(2*arctan(a*x))-1)/arctan(a*x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{2 \,{\left (a^{3} c^{2} x^{2} + a c^{2}\right )} \arctan \left (a x\right ) \int \frac{x}{{\left (a^{5} c^{2} x^{4} + 2 \, a^{3} c^{2} x^{2} + a c^{2}\right )} \arctan \left (a x\right )}\,{d x} - x^{2}}{{\left (a^{3} c^{2} x^{2} + a c^{2}\right )} \arctan \left (a x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a^2*c*x^2+c)^2/arctan(a*x)^2,x, algorithm="maxima")

[Out]

(4*(a^3*c^2*x^2 + a*c^2)*arctan(a*x)*integrate(1/2*x/((a^5*c^2*x^4 + 2*a^3*c^2*x^2 + a*c^2)*arctan(a*x)), x) -
x^2)/((a^3*c^2*x^2 + a*c^2)*arctan(a*x))

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Fricas [C]  time = 1.97127, size = 301, normalized size = 7. \begin{align*} -\frac{2 \, a^{2} x^{2} -{\left (i \, a^{2} x^{2} + i\right )} \arctan \left (a x\right ) \logintegral \left (-\frac{a^{2} x^{2} + 2 i \, a x - 1}{a^{2} x^{2} + 1}\right ) -{\left (-i \, a^{2} x^{2} - i\right )} \arctan \left (a x\right ) \logintegral \left (-\frac{a^{2} x^{2} - 2 i \, a x - 1}{a^{2} x^{2} + 1}\right )}{2 \,{\left (a^{5} c^{2} x^{2} + a^{3} c^{2}\right )} \arctan \left (a x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a^2*c*x^2+c)^2/arctan(a*x)^2,x, algorithm="fricas")

[Out]

-1/2*(2*a^2*x^2 - (I*a^2*x^2 + I)*arctan(a*x)*log_integral(-(a^2*x^2 + 2*I*a*x - 1)/(a^2*x^2 + 1)) - (-I*a^2*x
^2 - I)*arctan(a*x)*log_integral(-(a^2*x^2 - 2*I*a*x - 1)/(a^2*x^2 + 1)))/((a^5*c^2*x^2 + a^3*c^2)*arctan(a*x)
)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{x^{2}}{a^{4} x^{4} \operatorname{atan}^{2}{\left (a x \right )} + 2 a^{2} x^{2} \operatorname{atan}^{2}{\left (a x \right )} + \operatorname{atan}^{2}{\left (a x \right )}}\, dx}{c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(a**2*c*x**2+c)**2/atan(a*x)**2,x)

[Out]

Integral(x**2/(a**4*x**4*atan(a*x)**2 + 2*a**2*x**2*atan(a*x)**2 + atan(a*x)**2), x)/c**2

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{{\left (a^{2} c x^{2} + c\right )}^{2} \arctan \left (a x\right )^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a^2*c*x^2+c)^2/arctan(a*x)^2,x, algorithm="giac")

[Out]

integrate(x^2/((a^2*c*x^2 + c)^2*arctan(a*x)^2), x)